Construction of global optimization constrained NLP test cases from unconstrained problems

This paper presents a novel construction technique for constrained nonconvex Nonlinear Programming Problem (NLP) test cases, derived from the evaluation tree structure of standardized bound constrained problems for which the global solution is known. It is demonstrated in a step-by-step procedure how first an equality constrained problem can be derived from an unconstrained one, with bounds imposed on all variables, using the Directed Acyclic Graph (DAG) of the unconstrained objective function and the use of interval arithmetic to derive bounds for the new variables introduced. An advantage of the proposed methodology is that several standard unconstrained global optimization test cases can be constructed for varying number of optimization variables, thus leading to adjustable size derived NLP's. Further to this in a second step it is demonstrated how any subset of the equalities derived can be relaxed into inequalities giving an equivalent optimization problem. Finally, in a third step it is demonstrated how, by reducing the number of equality constraints derived, it is possible to obtain more complex expressions in the constraints and objective function. The methodology is highlighted throughout by motivating examples and a sample code in Mathematica™ is provided in the Appendix